# American Institute of Mathematical Sciences

June  2019, 24(6): 2901-2922. doi: 10.3934/dcdsb.2018291

## Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems

 Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, China

* Corresponding author: Jingliang Lv

Received  August 2018 Published  October 2018

Two n-species stochastic type K monotone Lotka-Volterra systems are proposed and investigated. For non-autonomous system, we show that there is a unique positive solution to the model for any positive initial value. Moreover, sufficient conditions for stochastic permanence and global attractivity are established. For autonomous system, we prove that for each species, there is a constant which can be represented by the coefficients of the system. If the constant equals 1, then the corresponding species will be nonpersistent on average. To illustrate the theoretical results, the corresponding numerical simulations are also given.

Citation: Dejun Fan, Xiaoyu Yi, Ling Xia, Jingliang Lv. Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2901-2922. doi: 10.3934/dcdsb.2018291
##### References:
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Soc., 354 (2002), 3535-3554. doi: 10.1090/S0002-9947-02-03032-5. Google Scholar [17] X. Liang and J. Jiang, The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka-Volterra systems, Nonlinearity, 16 (2003), 785-801. doi: 10.1088/0951-7715/16/3/301. Google Scholar [18] M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457. doi: 10.1016/j.jmaa.2010.09.058. Google Scholar [19] M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495-2522. doi: 10.3934/dcds.2013.33.2495. Google Scholar [20] M. Liu, K. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969-2012. doi: 10.1007/s11538-010-9569-5. Google Scholar [21] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994. Google Scholar [22] X. Mao and C. Yuan, Stochastical Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. Google Scholar [23] S. Smale, On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7. doi: 10.1007/BF00307854. Google Scholar [24] H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856-874. doi: 10.1137/0146052. Google Scholar [25] H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Diff. Eqns., 176 (2001), 195-222. doi: 10.1006/jdeq.2001.3981. Google Scholar [26] P. Tak$\acute{a}\check{c}$, Convergence to equilibrium on invariant $d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244. doi: 10.1016/0022-247X(90)90040-M. Google Scholar [27] P. Tak$\acute{a}\check{c}$, Domains of attraction of generic omega-limit sets for strongly monotone discrete-time semigroups, J. Reine. Angew. Math., 423 (1992), 101-173. doi: 10.1515/crll.1992.423.101. Google Scholar [28] Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol., 10 (1980), 401-415. doi: 10.1007/BF00276098. Google Scholar [29] Y. Takeuchi, N. Adachi and H. Tokumaru, Global stability of ecosystems of the generalized Volterra type, Math. Biosci., 42 (1978), 119-136. doi: 10.1016/0025-5564(78)90010-X. Google Scholar [30] C. C. Travis and W. M. Post, Dynamics and comparative statics of mutualistic communities, J. Theor. Biol., 78 (1979), 553-571. doi: 10.1016/0022-5193(79)90190-5. Google Scholar [31] C. Tu and J. Jiang, The coexistence of a community of species with limited competition, J. Math. Anal. Appl., 217 (1998), 233-245. doi: 10.1006/jmaa.1997.5711. Google Scholar [32] C. Tu and J. Jiang, The necessary and sufficient conditions for the global stability of type-K Lotka-Volterra system, Proc. Am. Math. Soc., 127 (1999), 3181-3186. doi: 10.1090/S0002-9939-99-05077-7. Google Scholar [33] C. Tu and J. Jiang, Global stability and permanence for a class of type-K monotone systems, SIAM J.Math. Anal., 30 (1999), 360-378. doi: 10.1137/S0036141097325290. Google Scholar [34] Y. Wang and J. Jiang, The long-run behavior of periodic competitive Kolmogorov systems, Nonlinear Anal.: Real World Appl., 3 (2002), 471-485. doi: 10.1016/S1468-1218(01)00034-7. Google Scholar [35] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplices for the discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6. Google Scholar [36] F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275-288. doi: 10.3934/dcdsb.2010.14.275. Google Scholar [37] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158. Google Scholar

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##### References:
 [1] X. Bai and J. Jiang, Comparison theorems for neutral stochastic functional differential equations, J. Diff. Equ., 260 (2016), 7250-7277. doi: 10.1016/j.jde.2016.01.027. Google Scholar [2] X. Bai and J. Jiang, Comparison theorem for stochastic functional differential equations and applications, J. Dyn. Diff. Equ., 29 (2017), 1-24. doi: 10.1007/s10884-014-9406-x. Google Scholar [3] I. Barbalat, Systems dequations differentielles d'osci d'oscillations nonlineaires, Rev. Math Pures. Appl., 4 (1959), 267-270. Google Scholar [4] A. Berman and R. J. Plemmons, Nonnegative Matrices in Mathematical Sciences, Academic Press, New York, 1979. Google Scholar [5] L. Chen and J. Jiang, Stochastic epidemic models driven by stochastic algorithms with constant step, Discrete Contin. Dyn. Syst. Ser. B, 21 (2017), 721-736. doi: 10.3934/dcdsb.2016.21.721. Google Scholar [6] S. Cheng, Stochastic population systems, Stoch. Anal. Appl., 27 (2009), 854-874. doi: 10.1080/07362990902844348. Google Scholar [7] T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339. doi: 10.1007/BF00275641. Google Scholar [8] M. W. Hirsch, Systems of differential equations which are competitive or cooperative Ⅰ: Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013. Google Scholar [9] M. W. Hirsch, Systems of differential equations that are competitive or cooperative Ⅱ: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439. doi: 10.1137/0516030. Google Scholar [10] M. W. Hirsch, Systems of differential equations which are competitive or cooperative Ⅲ: Competing species, Nonlinearity, 1 (1988), 51-71. doi: 10.1088/0951-7715/1/1/003. Google Scholar [11] M. W. Hirsch, System of differential equations that are competitive or cooperative. Ⅳ: Structural stability in three-dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234. doi: 10.1137/0521067. Google Scholar [12] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus: 2nd Edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-1-4612-0949-2. Google Scholar [13] X. Li, A. Gray, D. Jiang and X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28. doi: 10.1016/j.jmaa.2010.10.053. Google Scholar [14] X. Li, D. Jiang and X. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021. Google Scholar [15] X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation, Discrete. Contin. Dyn. Syst., 24 (2009), 523-545. doi: 10.3934/dcds.2009.24.523. Google Scholar [16] X. Liang and J. Jiang, On the finite dimensional dynamical systems with limited competition, Trans. Am. Math. Soc., 354 (2002), 3535-3554. doi: 10.1090/S0002-9947-02-03032-5. Google Scholar [17] X. Liang and J. Jiang, The dynamical behaviour of type-K competitive Kolmogorov systems and its application to three-dimensional type-K competitive Lotka-Volterra systems, Nonlinearity, 16 (2003), 785-801. doi: 10.1088/0951-7715/16/3/301. Google Scholar [18] M. Liu and K. Wang, Persistence and extinction in stochastic non-autonomous logistic systems, J. Math. Anal. Appl., 375 (2011), 443-457. doi: 10.1016/j.jmaa.2010.09.058. Google Scholar [19] M. Liu and K. Wang, Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations, Discrete Contin. Dyn. Syst., 33 (2013), 2495-2522. doi: 10.3934/dcds.2013.33.2495. Google Scholar [20] M. Liu, K. Wang and Q. Wu, Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73 (2011), 1969-2012. doi: 10.1007/s11538-010-9569-5. Google Scholar [21] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994. Google Scholar [22] X. Mao and C. Yuan, Stochastical Differential Equations with Markovian Switching, Imperial College Press, London, 2006. doi: 10.1142/p473. Google Scholar [23] S. Smale, On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7. doi: 10.1007/BF00307854. Google Scholar [24] H. L. Smith, Competing subcommunities of mutualists and a generalized Kamke theorem, SIAM J. Appl. Math., 46 (1986), 856-874. doi: 10.1137/0146052. Google Scholar [25] H. L. Smith and H. R. Thieme, Stable coexistence and bi-stability for competitive systems on ordered Banach spaces, J. Diff. Eqns., 176 (2001), 195-222. doi: 10.1006/jdeq.2001.3981. Google Scholar [26] P. Tak$\acute{a}\check{c}$, Convergence to equilibrium on invariant $d$-hypersurfaces for strongly increasing discrete-time semigroups, J. Math. Anal. Appl., 148 (1990), 223-244. doi: 10.1016/0022-247X(90)90040-M. Google Scholar [27] P. Tak$\acute{a}\check{c}$, Domains of attraction of generic omega-limit sets for strongly monotone discrete-time semigroups, J. Reine. Angew. Math., 423 (1992), 101-173. doi: 10.1515/crll.1992.423.101. Google Scholar [28] Y. Takeuchi and N. Adachi, The existence of globally stable equilibria of ecosystems of the generalized Volterra type, J. Math. Biol., 10 (1980), 401-415. doi: 10.1007/BF00276098. Google Scholar [29] Y. Takeuchi, N. Adachi and H. Tokumaru, Global stability of ecosystems of the generalized Volterra type, Math. Biosci., 42 (1978), 119-136. doi: 10.1016/0025-5564(78)90010-X. Google Scholar [30] C. C. Travis and W. M. Post, Dynamics and comparative statics of mutualistic communities, J. Theor. Biol., 78 (1979), 553-571. doi: 10.1016/0022-5193(79)90190-5. Google Scholar [31] C. Tu and J. Jiang, The coexistence of a community of species with limited competition, J. Math. Anal. Appl., 217 (1998), 233-245. doi: 10.1006/jmaa.1997.5711. Google Scholar [32] C. Tu and J. Jiang, The necessary and sufficient conditions for the global stability of type-K Lotka-Volterra system, Proc. Am. Math. Soc., 127 (1999), 3181-3186. doi: 10.1090/S0002-9939-99-05077-7. Google Scholar [33] C. Tu and J. Jiang, Global stability and permanence for a class of type-K monotone systems, SIAM J.Math. Anal., 30 (1999), 360-378. doi: 10.1137/S0036141097325290. Google Scholar [34] Y. Wang and J. Jiang, The long-run behavior of periodic competitive Kolmogorov systems, Nonlinear Anal.: Real World Appl., 3 (2002), 471-485. doi: 10.1016/S1468-1218(01)00034-7. Google Scholar [35] Y. Wang and J. Jiang, Uniqueness and attractivity of the carrying simplices for the discrete-time competitive dynamical systems, J. Differential Equations, 186 (2002), 611-632. doi: 10.1016/S0022-0396(02)00025-6. Google Scholar [36] F. Wu and Y. Hu, Stochastic Lotka-Volterra system with unbounded distributed delay, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 275-288. doi: 10.3934/dcdsb.2010.14.275. Google Scholar [37] M. L. Zeeman, Hopf bifurcations in competitive three-dimensional Lotka-Volterra systems, Dyn. Stability Systems, 8 (1993), 189-217. doi: 10.1080/02681119308806158. Google Scholar
Solutions of system (16) for $r_1 = 0.055,~r_2 = 0.045,~r_3 = 0.035,~a_{11} = 0.01,~a_{12} = -0.05,~a_{13} = 0.01,~a_{21} = -0.05$, $a_{22} = 0.1,~a_{23} = 0.01,~a_{31} = 0.01,~a_{32} = 0.01,~a_{33} = 0.1.$ The horizontal axis represents the time $t$. (a) is with $\sigma_1 = 0.3,~\sigma_2 = 0.3606,~\sigma_3 = 0.3243$; (b) is with $~\sigma_1 = 0.3,~\sigma_2 = 0.4359,~\sigma_3 = 0.3243$; (c) is with $\sigma_1 = 0.35,~\sigma_2 = 0.4359,~\sigma_3 = 0.2646$.
Solutions of system (16) for $r_1 = 0.55,~r_2 = 0.24,~r_3 = 0.36,~a_{11} = 0.095,~a_{12} = -0.05,~a_{13} = 0.01,~a_{21} = -0.05,$ $a_{22} = 0.0095,~a_{23} = 0.01,~a_{31} = 0.01,~a_{32} = 0.01,~a_{33} = 0.1,~\sigma_1 = 0.2,~\sigma_2 = 0.1612,~\sigma_3 = 0.1732.$ The horizontal axis represents the time $t$.
Solutions of system (16) for $r_1 = 0.55,~r_2 = 0.24,~r_3 = 0.36,~a_{11} = 0.095,~a_{12} = -0.05,~a_{13} = 0.01,~a_{21} = -0.05,$ $~a_{22} = 0.0095,~a_{23} = 0.01,~a_{31} = 0.01,~a_{32} = 0.01,~a_{33} = 0.1,~\sigma_1 =$$0.2,~\sigma_2 = 0.1612,~\sigma_3 = 0.1732,x_1(0) = 10.3,~y_1(0) = 10.2,~x_2(0) =$ $7.5,~y_2(0) = 7.3,~x_3(0) = 5.2,~y_3(0) = 5.1.$ The horizontal axis represents the time $t$.
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